An industrial engineer is evaluating the tensile strength of two specific grades of composite fiber braids, Grade X and Grade Y. Random samples are extracted from production and their strengths, s N/mm2s\text{ N/mm}^2s N/mm2, are recorded. The findings are summarized in the table below.
| Number in sample | Sample mean sˉ\bar{s}sˉ | ∑s2\sum s^2∑s2 | |
|---|---|---|---|
| Grade X | 12 | 85.5 | 87834.0 |
| Grade Y | 15 | 92.2 | 127625.4 |
You may assume that the samples are drawn from independent normal distributions with a common population variance.
The engineer claims that the mean tensile strength of Grade Y fiber is 5 N/mm2\text{N/mm}^2N/mm2 greater than the mean tensile strength of Grade X fiber.
Test the engineer's claim at the 5% level of significance.
Given that the true population variance for both grades is actually 10 (N/mm2)210\text{ (N/mm}^2)^210 (N/mm2)2,
(i) show that when samples of size 12 and 15 are used with a 5% level of significance, the engineer's claim is accepted if 2.60<SˉY−SˉX<7.402.60 < \bar{S}_Y - \bar{S}_X < 7.402.60<SˉY−SˉX<7.40.
(ii) Hence find the probability of a Type II error for this test if, in fact, the true mean tensile strength of Grade Y is 8 N/mm2\text{N/mm}^2N/mm2 greater than the mean tensile strength of Grade X.